Lecture 10 - Two-sided non-compliance
D ~ Z)Y ~ D_hat)estimatr:::iv_robust(Y ~ D | Z, data = data)Source: Alves (2022)
More formally, we have the following:
\(Z_i\) is the treatment assignment, \(D_i\) is the treatment receipt, and \(Y_i\) is the outcome
Compliers: \(D_i(1) = 1\) and \(D_i(0) = 0\). Similarly, \(D_i(1) \gt D_i(0)\)
Never-takers: \(D_i(1) = 0\) and \(D_i(0) = 0\)
Always-takers: \(D_i(1) = 1\) and \(D_i(0) = 1\)
Defiers: \(D_i(1) = 0\) and \(D_i(0) = 1\). These are usually rare, though
Connections between observed data and compliance types:
| \(Z_i = 0\) | \(Z_i = 1\) | |
|---|---|---|
| \(D_i = 0\) | Never-taker or Complier | Never-taker or Defier |
| \(D_i = 1\) | Always-taker or Defier | Always-taker or Complier |
Let \(\pi_{AT}\), \(\pi_{NT}\), \(\pi_{C}\), \(\pi_{D}\) denote proportions of Always-Takers, Never-Takers, Compliers, and Defiers
Formulas to estimate these proportions:
Intent-to-treat effect on outcome (\(ITT_Y\)) can be decomposed:
\[ ITT_Y = ITT_{Y,co} \pi_{co} + \underbrace{ITT_{Y,at} \pi_{at}}_{=0} + \underbrace{ITT_{Y,nt} \pi_{nt}}_{=0} + \underbrace{ITT_{Y,de} \pi_{de}}_{=0 \text{ (mono)}} \]
Under Exclusion Restriction (ER) and Monotonicity (mono) assumptions, ITT simplifies to:
\[ ITT_Y = ITT_{co} \pi_{co} \]
Same identification result:
\[ \tau_{LATE} = \frac{ITT_Y}{ITT_D} \]